Unit 7, Lesson 2
Inscribed Angles
Geometry
2.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that two chords form an angle that’s not a central angle, which will be useful when students investigate properties of inscribed angles in a subsequent activity. While students may notice and wonder many things about these images, the fact that the inscribed angle is formed by two chords and its vertex is on the circle is the most important discussion point.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Launch
Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Student Response
Loading...
Activity Synthesis
The goal of this discussion is for students to learn the definition of “inscribed angle.” Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
Tell students that an inscribed angle in a circle is an angle formed by two chords that share an endpoint. An inscribed angle’s vertex lies on the circle. The other two endpoints of the chords define an arc. When we talk about the arc that an inscribed angle defines, we are referring to the arc that does not pass through the vertex of the inscribed angle.
Lesson Synthesis
In this lesson, students conjectured that the measure of an inscribed angle is half the measure of the central angle that defines the same arc. Tell students that in the unit “Coordinate Geometry,” they investigated a special case of this theorem that they will revisit now.
Invite students to use a compass to draw a circle. Then they should place an index card so a corner lies on the circle and the sides of the card form an inscribed angle. Students should trace the index card to draw this angle, then connect the points where the chords intersect the circle. Ask students what they notice about this third line, and why does this happen? (It is a diameter. It goes through the center of the circle. The index card created a 90-degree inscribed angle, so the central angle must measure twice that, or 180 degrees. A central angle that is a straight line is a diameter.)
Remind students that in the unit “Coordinate Geometry,” we started the other way around, by drawing a diameter. Ask students, “What does the Inscribed Angle Theorem tell us about any inscribed angle that defines a 180-degree arc?” (It must measure 90 degrees.)
Student Lesson Summary
We have discussed central angles such as angle . Another kind of angle in a circle is an inscribed angle, or an angle formed by two chords that share an endpoint. In the image, angle is an inscribed angle.
It looks as though the inscribed angle is smaller than the central angle that defines the same arc. In fact, the measure of an inscribed angle is always exactly half the measure of the associated central angle. For example, if the central angle measures 50 degrees, the inscribed angle must measure 25 degrees, even if we move point along the circumference (without going past or ). This also means that all inscribed angles that define the same arc are congruent.